scholarly journals Some nodal properties of the quantum harmonic oscillator and other Schrödinger operators in ℝ²

2017 ◽  
pp. 87-116 ◽  
Author(s):  
Pierre Bérard ◽  
Bernard Helffer
Keyword(s):  
Harmonic Oscillator ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Quantum Harmonic Oscillator ◽  
Nodal Properties

scholarly journals Asymptotic eigenfunctions for Schrödinger operators on a vector bundle

2020 ◽  
Vol 32 (07) ◽  
pp. 2050020
Author(s):  
Matthias Ludewig ◽  
Elke Rosenberger
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Potential Energy ◽  
Harmonic Oscillator ◽  
Tangent Space ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Smooth Functions ◽  
Energy Formula ◽  
Laplace Type

In the limit [Formula: see text], we analyze a class of Schrödinger operators [Formula: see text] acting on sections of a vector bundle [Formula: see text] over a Riemannian manifold [Formula: see text] where [Formula: see text] is a Laplace type operator, [Formula: see text] is an endomorphism field and the potential energy [Formula: see text] has a non-degenerate minimum at some point [Formula: see text]. We construct quasimodes of WKB-type near [Formula: see text] for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions of the associated harmonic oscillator [Formula: see text] at [Formula: see text], acting on smooth functions on the tangent space.

scholarly journals Structure of the Spectra and Resonances of Schrödinger Operators

2020 ◽  
Vol 12 (1) ◽  
pp. 83-102
Author(s):  
Tahar Bouguetaia ◽  
Bekkai Messirdi
Keyword(s):  
Hydrogen Atom ◽  
Harmonic Oscillator ◽  
Schrödinger Operator ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Mathematical Physics ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Operator Spectrum

The main goal of this paper is to study the spectrum and resonances of several classes of Schrödinger operators. Two important examples occurring in mathematical physics are discussed: harmonic oscillator and Hamiltonian of hydrogen atom. Keywords: Schrödinger operator, Spectrum, Periodic potential, Resonances.

scholarly journals Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Luca Fresta
Keyword(s):  
Density Of States ◽  
Local Density ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Schrodinger Operators ◽  
Local Density Of States ◽  
Weak Disorder ◽  
Cluster Expansions ◽  
Strong Disorder ◽  
Lifshitz Tail

AbstractWe study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.

Sign in / Sign up

Export Citation Format

Share Document